The spin quantum number of particles with fractional spin is neither an integer nor a half-integer. Quasiparticles knownas anyons are the most familiar examples of such particles. However, despite the considerable interest in anyons dueto their potential applications in quantum computation, a systematic analysis within the framework of quantummechanics addressing the fundamental question of whether elementary particles with fractional spin can exist is stilllacking. Two major obstacles are the requirement that spin operators satisfy an su(2) algebra and the spin-statisticstheorem. Focusing on spin-1/4 particles, we show that the first obstacle can be overcome by assuming that their spinoperators obey a deformed su(2) algebra with unequal structure constants. This symmetry breaking leads to a uniquerotational symmetry axis and imposes that the two particles forming a bound state have aligned symmetry axes andidentical spin projections. The second obstacle can be overcome by assuming that the binding interaction acts betweenparticles of different species, aligning their spins and confining their relative motion to the plane perpendicular to thesymmetry axis. Under these conditions, the braid operators describing the interchange of two spin-1/4 particles can berepresented in terms of the spin-component operator associated with the symmetry axis, closely paralleling theconventional spin-statistics connection. These results establish a framework in which spin-1/4 particles can beconsistently incorporated into quantum mechanics. Finally, we discuss the potential relevance of the proposedframework to theories beyond the Standard Model in which quarks are regarded as composite particles.
The spin quantum number of particles with fractional spin is neither an integer nor a half-integer. Quasiparticles knownas anyons are the most familiar examples of such particles. However, despite the considerable interest in anyons dueto their potential applications in quantum computation, a systematic analysis within the framework of quantummechanics addressing the fundamental question of whether elementary particles with fractional spin can exist is stilllacking. Two major obstacles are the requirement that spin operators satisfy an su(2) algebra and the spin-statisticstheorem. Focusing on spin-1/4 particles, we show that the first obstacle can be overcome by assuming that their spinoperators obey a deformed su(2) algebra with unequal structure constants. This symmetry breaking leads to a uniquerotational symmetry axis and imposes that the two particles forming a bound state have aligned symmetry axes andidentical spin projections. The second obstacle can be overcome by assuming that the binding interaction acts betweenparticles of different species, aligning their spins and confining their relative motion to the plane perpendicular to thesymmetry axis. Under these conditions, the braid operators describing the interchange of two spin-1/4 particles can berepresented in terms of the spin-component operator associated with the symmetry axis, closely paralleling theconventional spin-statistics connection. These results establish a framework in which spin-1/4 particles can beconsistently incorporated into quantum mechanics. Finally, we discuss the potential relevance of the proposedframework to theories beyond the Standard Model in which quarks are regarded as composite particles.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ = 0, π correspond to bosons and fermions, respectively; at intermediate values, we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve nonabelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of nonabelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern–Simons terms. The crossings of one-dimensional anyons on a ring are unidirectional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently, the anyon behavior predicted for quasiparticles in the ν = 1/3 fractional quantum Hall state has been observed in both scattering and interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ=0,π correspond to bosons and fermions respectively; at intermediate values we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve non-abelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of non-abelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern--Simons terms. The crossings of one-dimensional anyons on a ring are uni-directional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently the anyon behavior predicted for quasiparticles in the ν=1/3 fractional quantum Hall state has been observed both in scattering and in interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
In our previous papers it has been shown that quantum theory based on a Galois field (GFQT) possesses a new symmetry between particles and antiparticles, and for massless particles this symmetry (called the AB one) is compatible with all representation operators of the symmetry algebra. In the present paper, it is shown that the AB symmetry is compatible with all representation operators of the symmetry algebra in the general case. As a consequence of simple arithmetical considerations, this symmetry is compatible with the vacuum condition, only for particles with half-integer spin (in usual units). If the AB symmetry is combined with the spin-statistics theorem, one arrives at the following conclusions: in quantum theory based on a Galois field i) any neutral particle can be only composite but not elementary (this property has been proved for the massless case in hep-th/0207192); ii) any interaction can involve only an even number of creation/annihilation operators.
The angular momentum quantum number L of spherical harmonic Y_l_,_m based on an associated Legendre polynomial is nonnegative integer 0 1 2 ... and must never be a fraction. But the study in this paper found that the quantum number L corresponding to other series of solutions of the associated Legendre equation should be fractions. This paper not only proposed the spherical harmonics Y for L=1/2, but also spherical harmonics Y for L = 1/n = 1/3 1/4 1/5 ... In addition to the spin s=1/2 of electron-like particles, the fractional spin such as 1/3, 1/4, 1/5,...were boldly speculated to be verified in this paper. Setting the spin of a particle with only two spin components of up and down to s = 1/2 is not necessarily correct. Based on the symmetry of the plots of Ys, three different spin classes of particles are predicted. The first class of particles s = 1/2 1/6 ... resembles electrons, particles with the parallel spins tend to move away from each other, and the second class of particles s = 1/3 1/5 ... does not repel each other regardless of whether their spins are parallel or not, and the third class of particles s = 1/4 1/8 ... always repels each other and tend to move away regardless of whether their spins are parallel or not. The use of fractional spin can be well illustrated for electrons and protons as reported in the literature. This view may be important for the study of quantum mechanics and elementary practices.
This chapter gives a brief introduction to rotations in two-dimensional space and finds that rotations in two-dimensional space have some special properties. In particular, spin and statistics have peculiarities not possible in three- or higher-dimensional space—spin is not quantized in integer or half-integer units of h and hence the particles are not necessarily bosons or fermions but may obey any statistics and were thus called anyons by Wilczek. In fact, the concept of fractional statistics had been already put forward by Leinaas and Myrheim in the late 1970s. They found that the origin of the fractional statistics lies in the peculiar topological properties of the configuration space of many identical particles. This space is doubly connected in three or more dimensions but is multiply connected in two dimensions. Later Goldin, Menikoff and Sharp reached a similar conclusion using a completely different method based on the rigorous study of the unitary representations of current algebra and diffeomorphism groups.
The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.